*PM : derivation of directional derivative, id = 7267-- WP guess : derivation of directional derivative-- Status:
22.
*PM : derivation of directional derivative, id = 7267-- WP guess : derivation of directional derivative-- Status:
23.
This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued.
24.
It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives.
25.
Therefore, any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative.
26.
Therefore, any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative.
27.
The first approach is to examine what is required for a generalization of the directional derivative to " behave well " under coordinate transitions.
28.
The gradient of is defined as the unique vector field whose dot product with any vector at each point is the directional derivative of along.
29.
If is differentiable, then the dot product of the gradient at a point with a vector gives the directional derivative of at in the direction.
30.
By the Cauchy-Riemann equations these tangential derivative are up to a sign the directional derivative in the direction of the normal to the boundary.
